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URL: https://oeis.org/A353709

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A353709
a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations and also a(n) and a(n-1) have no common 1-bits in their binary representations.
21
0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6, 40, 64, 5, 10, 48, 65, 14, 128, 33, 18, 68, 9, 34, 20, 72, 35, 132, 24, 66, 36, 25, 130, 96, 13, 144, 98, 256, 21, 42, 192, 257, 22, 104, 129, 258, 28, 97, 384, 26, 37, 320, 136, 7, 80, 160, 11, 84, 288, 131, 76, 272, 161, 70, 264, 49, 134, 328, 512, 19, 44, 448, 513, 30, 224, 768, 15, 112, 640, 259, 52, 200, 514, 53
OFFSET
0,3
COMMENTS
A set-theory analog of A084937.
Conjecture: This is a permutation of the nonnegative numbers.
MAPLE
read(transforms) : # ANDnos def'd here
A353709 := proc(n)
option remember;
local c, i, known ;
if n <= 2 then
n;
else
for c from 1 do
known := false ;
for i from 1 to n-1 do
if procname(i) = c then
known := true;
break ;
end if;
end do:
if not known and ANDnos(c, procname(n-2)) = 0 and ANDnos(c, procname(n-1)) = 0 then
return c;
end if;
end do:
end if;
end proc: # Following R. J. Mathar's program for A109812.
[seq(A353709(n), n=0..256)] ;
# Alternative:
b:= proc() false end: t:= 2:
a:= proc(n) option remember; global t; local k; if n<2 then n
else for k from t while b(k) or Bits[And](k, a(n-2))>0
or Bits[And](k, a(n-1))>0 do od; b(k):=true;
while b(t) do t:=t+1 od; k fi
end:
seq(a(n), n=0..100); # Alois P. Heinz, May 06 2022
MATHEMATICA
nn = 83; c[_] = -1; a[0] = c[0] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[a[n - 1], k] == 0, BitAnd[a[n - 2], k] == 0], k++]; Set[{a[n], c[k]}, {k, n}]; If[k == u, While[c[u] > -1, u++]], {n, 2, nn}]; Array[a, nn+1, 0] (* Michael De Vlieger, May 06 2022 *)
PROG
(Python)
from itertools import count, islice
def A353709_gen(): # generator of terms
s, a, b, c, ab = {0, 1}, 0, 1, 2, 1
yield from (0, 1)
while True:
for n in count(c):
if not (n & ab or n in s):
yield n
a, b = b, n
ab = a|b
s.add(n)
while c in s:
c += 1
break
A353709_list = list(islice(A353709_gen(), 20)) # Chai Wah Wu, May 07 2022
CROSSREFS
Cf. A084937 (number theory analog), A109812, A121216, A353405 (powers of 2), A353708, A353710, A353715 and A353716 (a(n)+a(n+1)), A353717 (inverse), A353718, A353719 (primes), A353720 and A353721 (Records).
For the numbers that are the slowest to appear see A353723 and A353722.
Sequence in context: A242365 A119436 A277695 * A317503 A243505 A386248
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, May 06 2022
STATUS
approved